Handling Uncertainty in Complex Systems · Handling Uncertainty in Complex Systems Dr. Serena Doria Department of Engineering and Geology, University G. d’Annunzio The aim of the

Handling Uncertainty in Complex Systems – Serena Doria -University G. d’Annunzio Chieti - Pescara

DOCTORAL COURSE IN EARTH SYSTEMS AND BUILT ENVIRONMENTS

University Gabriele d’Annunzio Chieti-Pescara - Italy

Handling Uncertainty in Complex Systems

Dr. Serena Doria Department of Engineering and Geology,

University G. d’Annunzio

The aim of the course is to introduce some new results proposed in literature to represent and handling uncertainty in complex systems.



Program -Complex systems and complex decisions in the real world. Imprecise probability: a tool to represent partial knowledge.

-Iterated functions systems, attractors, fractal set: mathematical models to represent complex systems.

-A measure of the complexity of a set: the Hausdorff dimension; Hausdorff dimensional outer measures.

- A new model of upper and lower coherent conditional previsions based on Hausdorff outer measures.

- Coherent measures of risk defined by the Choquet integral with respect to Hausdorff outer measures: applications.



References 1. Billingsley, P. Probability and measure. Wiley, (1986) 2. Blackwell, D., Dubins L. On existence and non existence of proper, regular, conditional distributions. The annals of Probability, Vol. 3, No 5, 741-752 Wiley, (1975) 3. Choquet, G. Theory of capacities. Ann. Inst. Fourier, 5, 131-295, (1953) 4. de Finetti, B. Probability, Induction and Statistics, Wiley, New York, (1972) 5. de Finetti, B. Theory of Probability, Wiley, London, (1974) 6. Denneberg, D. Non-additive measure and integral. Kluwer Academic Publishers, (1994) 7. Doria, S. Probabilistic independence with respect to upper and lower conditional probabilities assigned by Hausdor_ outer and inner measures. International Journal of Ap- proximate Reasoning, 46, 617-635, (2007) 8. Doria, S. Convergences of random variables with respect to coherent upper probabilities de_ned by Hausdor_ outer measures, Advances in Soft Computing, 48, Springer, 281-288 (2008) 9. Doria, S. Coherent upper conditional previsions and their integral representation with respect to Hausdor_ outer measures Combining Soft Computing and Stats Methods, C.Borgelt et al. (Eds.) 209-216, (2010) 10. Doria, S. Coherent Upper and Lower Conditional Previsions De_ned by Hausdor_ Outer and Inner Measures in Modeling, Design, and Simulation Systems with Uncertain- ties, Mathematical Engineering, Springer, Volume 3, 175-195, (2011) 11. Doria, S. Characterization of a coherent upper conditional prevision as the Choquet integral with respect to its associated Hausdor_ outer measure, Annals of Operations Research, 195, 33-48, (2012) 12. Doria, S. Symmetric coherent upper conditional prevision by the Choquet integral with respect to Hausdorff outer measure, Annals of Operations Research, 229, 377-396, (2014) 13. Doria, S. Coherent conditional measures of risk de_ned by the Choquet integral with respect to Hausdorff outer measure and stochastic independence in risk management,



International Journal of approximate Reasoning, 65, 1-10, (2015) 14. Doria, S. On the disintegration property of a coherent upper conditional prevision by the Choquet integral with respect to its associated Hausdorff outer measure, Annals of Operations Research,Volume 256, Iussue 2, 253-269, (2017) 15. Di Cencio, A., Doria, S. Probabilistic Analysis of Suture Lines Developed in Am- monites: The Jurassic Examples of Hildocerataceae and Hammatocerataceae, Mathemat- ical Geosciences, Volume 49, Issue 6, 737-750, (2017) 16. Dubins, L.E. Finitely additive conditional probabilities, conglomerability and disinte- grations. The Annals of Probability, Vol. 3, 89-99, (1975) 17. Falconer, K.J. The geometry of fractals sets. Cambridge University Press, (1986) 18. Grabisch, M. Set Functions, Games and Capacities in Decision Making. Springer, (2016) 19. Regazzini, E. Finitely additive conditional probabilities, Rend. Sem. Mat. Fis., Milano, 55,69-89, (1985) 20. Regazzini, E. De Finetti's coherence and statistical inference, The Annals of Statistics, Vol. 15, No. 2, 845-864, (1987) 21. Rogers, C. A. Hausdorff measures. Cambridge University Press, (1970) 22. Srivastava, S. M. A Course on Borel Sets, Spring, (1998) 23. Troffaes M., de Cooman G. Lower Previsions, Wiley, (2014) 24. Walley, P. Coherent lower (and upper) probabilities, Statistics Research Report, Uni- versity of Warwick, (1981) 25. Walley, P. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, (1991)



Deterministic phenomena in which, starting from known initial conditions, we can determine the values of the quantities under examination and therefore the outcome of the phenomenon (e.g. a falling object) Random phenomena or well-defined phenomena whose outcome is not known until the phenomenon occurs (e.g. of a coin) Complex phenomena characterized by a non-linear response to perceived stimuli (e.g. sand castle, landslides, earthquakes, floods, chorus)



A complex decision is a decision where the best alternative cannot be obtained by a preference ordering represented by a linear functional

A random variable X is a bounded function from Ω to the real numbers ℝ. This function is called gamble in Walley (1991) or random quantity in de Finetti (1972, 1974).



A functional " defined on the class L(B) of all random variables defined on a non-empty set B is linear if for every #, % ∈ L(B) and for every ', ( ∈ ℛ

"*'# + (%, = '"*#, + ("*%,



A preference ordering on the class L(B) of all random variables defined on B is represented by a linear functional " (e.g. the weighted sum) if and only if

# ≻ % ⇔ "*#, > "*%, and

# ≃ % ⇔ "*#, = "*%, Nevertheless not all preference orderings can be represented by a linear functional



Example 1 Let Ω be a non-empty set, B = 234, 345 and let 6 be a probability measure defined on the field generated by B. Let consider the class 7 = 284, 894, 8:5 of bounded B-measurable random variables defined on B by

Random variables

34 39

84 0.3 0.3 89 0.7 0 8: 0 0.7



The preference ordering 84 ≻ 89 and 89 ≃ 8: cannot be represented by the linear functional

;*8, = < 6*3=

9

=>4,?=

since there exists no probability measure 6 such that the following system has solution: _



!84 ≻ 8984 ≃ 89

⇔ "0.36*34, + 0.36*39, > 0.76*34, + 06*39,0.76*34, + 06*39, = 06*34, + 0.76*39,

6*34, + 6*39, = 1

Remark If only the random variables 84 and 89 are considered we have that the preference 84 ≻ 89 can be represented by a linear functional, since

84 ≻ 89 ⇔ "0.36*34, + 0.36*39, > 0.76*34, + 06*39,

6*34, + 6*39, = 1



and the system has solutions: all pair (6*34,; 6*39,* with 6*34, < :

, and 6*39, = 1 − 6*34,. The previous example put in evidence the necessity to introduce non-linear functionals to represent preference orderings and to investigate equivalent random variables.



For B in B let X|B be the restriction to B of a random variable X defined on Ω and let sup X|B be the supreme value assumed by X on B. For B in B and X|B in K(B) a coherent upper conditional prevision .*8|3, is a real functional on K(B) such that the following conditions hold for every X|B and Y|B in K(B) and positive constant λ:



1. .*8|3, ≤ sup 8|3

2. .*48|3, = 4.*8|3, positive homogeneity

3. .*8 + 5|3, ≤ .*8|3, + .*5|3, subadditivity

4. .*3|3, = 1



If .*8|3, is a coherent upper conditional prevision on a linear space K(B) then its conjugate coherent lower conditional prevision is defined by .*8|3, = −.*−8|3, and

.*8|3, ≤ .*8|3, If for every X|B belonging to K(B) we have

.*8|3, = .*8|3, = .*8|3, then .*8|3, is called a coherent linear conditional prevision and it is a linear, positive functional on K(B).



For each X in K(B) let .*8|6) be the function defined on Ω by

.*8|6,*7, = .*8|3, if 7 ∈ 3.

.*8|6, is called a coherent upper conditional prevision and it is separately coherent if all the .*8|3, are separately coherent.





A necessary and sufficient condition for an upper prevision . to be coherent is to be the upper envelope of linear previsions, i.e. there exists a class M of linear previsions, defined on a same domain, such that

.*8, = sup2.*8,: . ∈ 95



Example 1 (continued) We consider the following class of probability measures 34 39

.4 1 0

.9 0 1

6 1 1 6 0 0



Let 6: be the coherent upper conditional probability defined by

6 *;, = <=?2.4*;,, .9*;,5 and let 6: be the coherent lower conditional probability defined by

6 *;, = <>?2.4*;,, .9*;,5



The coherent upper and lower conditional previsions can be represented as Choquet integral If the atoms 3= are enumerated so that ?= = 8*3=, are in in descending order, i.e. ?4 ≥ ?9 ≥ ⋯ ≥ ?B and ?BC4 = 0 the Choquet integral with respect to µ is given by

D E 8F6 = <*?=

B

=>4− ?=C4,6*G=,



where G= = 34 ∪ 39 ∪ … ∪ 3=. In Example 1 we have

.*84, = 0.3 and .*89, = .*8:, = 0 so that the preference ordering

84 ≻ 89 and 89 ≃ 8:. can be represented by the coherent lower prevision ..



The preference ordering cannot be represented by the coherent upper prevision . since

.*84, = 0.3 and .*89, = .*8:, = 0.7



A strict ordering, induced by a coherent lower conditional prevision .*⋅ |3, can be defined on the class of random variables belonging to L(B) ( Walley (Section 3.8.1)): Definition 1 We say that the alternative 8= is preferable to 8K given B with respect to .*⋅ |3,, i.e. 8= ≻∗ 8K given B if and only if

.(8= − 8KM3* > 0.



Some information can be lost when a strict preference order is defined by to .*⋅ |3, since . does not contain any information about which random variable, with .*8|3, = 0 are really desirable.



In Walley (section 3.9.4) the following definitions are given. Definition 2 Let K be a finite class of random variables. A random variable 8= in K is inadmissible in K given B if there is 8K in K such that 8K ≻∗ 8= with j≠i. Otherwise 8= is admissibile in K. Definition 2 An admissible random variable 8= in K is maximal in K given B under the coherent lower prevision .*⋅ |3, when 8= is admissible in K and



.(8= − 8KM3* ≥ 0 ∀8K in K.

If .*⋅ |3, is a linear prevision, the maximal random variable belonging to L(B) with respect to .*⋅ |3, is the admissible random variable 8= which satisfies

.*8=|3, ≥ .(8KM3* for all 8K in K.

Any alternative which maximizes .*8K|3, over 8K in K is called a Bayes alternative under .*⋅ |3,.



A Bayes random variable under a coherent lower conditional prevision is a random variable which is maximal under a linear prevision on the class of all random variables defined on B. Definition 3 An admissible random variable 8= is defined to be a Bayes random variable under a coherent lower prevision . when, for each 8K in K there is a linear prevision . ∈ 9*., such that .*8=|3, ≥ .(8KM3*∀8K in K.



If 8= is maximal under some . ∈ 9*., then

.*8=|3, ≥ .(8KM3* ∀8K in K so

.(8= − 8KM3* ≥ .(8= − 8KM3* = .*8=, − .*8K, ≥ 0

and 8= is maximal under .. So a Bayes random variable under a coherent lower prevision . is maximal under . but the converse is not true.

8 >O = 3=PQO R=?FS< T=R>=UVQ W?FQR . ⇒ 8 >O <=?><=V W?FQR .



Example 1 (continued)

Random variables

34 39

84 0.3 0.3 89 0.7 0 8: 0 0.7



We consider the following class of probability measures 34 39

.4 1 0

.9 0 1

6 1 1 6 0 0

We obtain .(8= − 8K* = D Z(8= − 8K*F6 < 0 for all >, [ ∈ 21,2,35 with > ≠ [

so all random variables 8= for all > ∈ 21,2,35 are admissible



and

.(8= − 8K* = D Z(8= − 8K*F6 ≥ 0 for all >, [ ∈ 21,2,35

so all random variables 8= for all > ∈ 21,2,35 are maximal. No random variable 8= for all > ∈ 21,2,35 is a Bayes random variable.






University G. d’Annunzio The aim of the course is to introduce some new results proposed in literature to represent and handling uncertainty in complex systems.



A new model of coherent upper and lower conditional previsions based on Hausdorff

outer measures.



A random variable X is a bounded function from Ω to the real numbers ℝ. This function is called gamble in Walley (1991) or random quantity in de Finetti (1972, 1974).



For B in B let X|B be the restriction to B of a random variable X defined on Ω and let sup X|B be the supreme value assumed by X on B. For B in B and X|B in K(B) a coherent upper conditional prevision "($|&) is a real functional on K(B) such that the following conditions hold for every X|B and Y|B in K(B) and positive constant λ:



1. "($|&) ≤ sup$|&

2. "(-$|&) = -"($|&) positive homogeneity

3. "($ + 0|&) ≤ "($|&) + "(0|&) subadditivity

4. "(&|&) = 1



If "($|&) is a coherent upper conditional prevision on a linear space K(B) then its conjugate coherent lower conditional prevision is defined by "($|&) = −"(−$|&) and

"($|&) ≤ "($|&) If for every X|B belonging to K(B) we have

"($|&) = "($|&) = "($|&)



then "($|&) is called a coherent linear conditional prevision and it is a linear, positive functional on K(B).

For each X in K(B) let "($|3) be the function defined on Ω by

"($|3)(4) = "($|&) if 4 ∈ &.

"($|3) is called a coherent upper conditional prevision and it is separately coherent if all the "($|&) are separately coherent.





A necessary and sufficient condition for an upper prevision " to be coherent is to be the upper envelope of linear previsions, i.e. there exists a class M of linear previsions, defined on a same domain, such that

"($) = sup6"($): " ∈ 89



A random variable X on Ω is B-measurable or measurable with respect to the partition B if it is constant on the atoms of the partition B (Walley 1991, p.291).

Necessary condition for coherence If for every & ∈ 3 P(X|B) are coherent linear conditional previsions and X is B-measurable then (Walley 1991, p. 292)

"($|3) = $



Motivations

Why the necessity to propose a new model of coherent upper conditional prevision "($|3) ?



Because separately coherent upper conditional prevision "($|3) cannot always be defined as extension of conditional expectation :($|;) of measurable random variables. In fact conditional expectation :($|;) defined by the Radon-Nikodym derivative, according to the axiomatic definition, may fail to be separately coherent.



It occurs because one of the defining properties of the Radon- Nikodym derivative, that is to be measurable with respect to the sigma-field of the conditioning events, contradicts a necessary condition for the coherence ("($|3) = $).



Linear coherent conditional prevision "($|3) can be compared with conditional expectation :($|;) if the partition B generates the sigma-field G (Koch, 1997, p. 262)

:($|;)(4) = "($|&) if 4 ∈ & with & ∈ 3



Definition of conditional expectation (Billingsley, 1986)

Let F and G be two sigma-fields of subsets of Ω with G ⊂ F and let X be an integrable, F-measurable random variable. Let P be a probability measure on F ; define a measure = on G by

=(>) = ? $ "! .

This function is unique up to a set of P-measure zero and it is a version of the conditional expectation value.



This measure is finite and absolutely continuous with respect to P (i.e. "(") = 0 ⟹ =(") = 0 ∀ " ∈ & ). Thus there exists a function, the Radon-Nikodym derivative, denoted by :($|;), G-measurable, integrable and satisfying the functional equation

? :($|;) " = ? $ "!! with G in ;.

This function is unique up to a set of P-measure zero and it is a version of the conditional expectation value.



If linear conditional prevision "($|3) is defined by the Radon-Nikodym derivative the necessary condition for coherence "($|3) = $ is not always satisfied.



Theorem 1 Let Ω = [0,1], let F be the Borel sigma-field of [0,1] and let P be the Lebesgue measure on F. Let G be a sub sigma-field properly contained in F and containing all singletons of [0,1]. Let B be the partition of all singletons of [0,1] and let X be the indicator function of an event A belonging to F-G. If we define the linear conditional prevision

"($|649) equal to the Radon-Nikodym derivative with probability 1, that is

"($|649) = :($|;) except on a set N of [0,1] of P-measure zero, then the conditional prevision "($|649) is not coherent.

(Theorem 1 S. Doria, Annals of Operation Research, Vol. 195, pp. 38-44, 2012)



If the equality "($|649) = :($|;) holds with probability 1, the linear conditional prevision "($|649) is different from X, the indicator function of A. In fact having fixed A in F-G the indicator function of A is not G-measurable. It occurs because for every Borel set C

$+,(-) = 64 ∈ Ω ∶ $(4) ∈ -9 =



=

/0102∅ if 0, 1 ∉ -" if 1 ∈ - and 0 ∉ -": if 0 ∈ - and 1 ∉ -Ω if 0, 1 ∈ -

and since A ∉ G then X is not G-measurable.



Example 1 (Billingsley, 1986; Seidenfeld et al. 2001) Let Ω = [0,1] F = Borel sigma-field of Ω, P = the Lebesgue measure on F

G = the sub sigma-field of sets that are either countable or co-countable B = the partition of all singletons of [0,1].



If the linear conditional prevision is equal, with probability 1, to conditional expectation defined by the Radon-Nikodym derivative, we have that

"($|3) = :($|;) = "($)

since the events in G have probability either 0 or 1.



So when X is the indicator function of an event A = [a,b] with 0 < a < b < 1 then

"($|3) = "(")

and it does not satisfy the necessary condition for coherence, that is

"($|649) = $ .



Evident from Theorem 1 and Example 1 is the necessity to introduce a new mathematical tool to define coherent conditional previsions.



The model For every & ∈ 3 denote by s the Hausdorff dimension of the conditioning event B and by ℎ< the Hausdorff s-dimensional outer measure.

ℎ< is called the Hausdorff outer measure associated with the coherent upper conditional prevision "($|&).





Theorem 2. Let L(B) be the class of all bounded random variables on B and let m be a 0-1-valued finitely additive, but not countably additive, probability on ℘(&) such that a different m is chosen for each B. Then for each & ∈ 3 the functional "($|&) defined on L(B) by

"($|&) = ,>?(@) ? $ ℎ<@ if 0 < ℎ<(&) < +∞

"($|&) = C($&) if ℎ<(&) = 0,+∞

is a coherent upper conditional prevision. (Theorem 2 , S. Doria, Annals of Operation Research, Vol. 195, pp.38-44, 2012)



The unconditional coherent upper prevision "($|Ω) is obtained as a particular case when the conditioning event is Ω.

Coherent upper conditional probabilities "("|&) are obtained when only 0-1 valued random variables are considered.

Theorem 3 Let m be a 0-1-valued finitely additive, but not countably additive, probability on ℘(&) such that a different m is chosen for each B. Then for each & ∈ 3 the function "($|&) defined on ℘(&) by



"("|&) = ℎ<("&)ℎ<(&) DE 0 < ℎ<(&) < +∞

and by

"("|&) = C("&) DE ℎ<(&) = 0,+∞ is a coherent upper conditional probability.



Main Results Let B be a set with positive and finite Hausdorff outer measure in its Hausdorff dimension. Denote by

F@∗ (") = "("|&) = ℎ<("&)ℎ<(&)

the coherent upper conditional probability defined on ℘(&).

From Theorem 2 we have that the coherent upper conditional prevision "(∙ |&) is a functional on I(&) with



values in ℝ and the coherent upper conditional probability F@∗ integral represents "($|&) since

"($|&) = J$ F@∗ =1

ℎ<(&) J $ ℎ<@



If the conditioning event has positive and finite Hausdorff outer measure in its

Hausdorff dimension and K(B) is a linear lattice of bounded random variables

containing all constants

Necessary and sufficient conditions for a coherent upper conditional prevision to be

uniquely represented as the Choquet integral with respest to the upper conditional probability defined by its associated Hausdorff outer measure are to be :

Monotone Comonotonically additive Submodular Continuous from

below






University G. d’Annunzio The aim of the course is to introduce some new results proposed in literature to represent and handling uncertainty in complex systems.



A new model of coherent upper and lower conditional previsions based on Hausdorff

outer measures.



Let B be a partition of Ω. For B in B let X|B be the restriction to B of a random variable X defined on Ω and let sup X|B be the supreme value assumed by X on B. For B in B and X|B in K(B) a coherent upper conditional prevision !(#|%) is a real functional on K(B) such that the following conditions hold for every X|B and Y|B in K(B) and positive constant λ:



1. !(#|%) ≤ sup #|%

2. !(,#|%) = ,!(#|%) positive homogeneity

3. !(# + /|%) ≤ !(#|%) + !(/|%) subadditivity

4. !(%|%) = 1



If !(#|%) is a coherent upper conditional prevision on a linear space K(B) then its conjugate coherent lower conditional prevision is defined by !(#|%) = −!(−#|%) and

!(#|%) ≤ !(#|%) If for every X|B belonging to K(B) we have

!(#|%) = !(#|%) = !(#|%) then !(#|%) is called a coherent linear conditional prevision and it is a linear, positive functional on K(B).



For each X in K(B) let !(#|2) be the function defined on Ω by

!(#|2)(3) = !(#|%) if 3 ∈ %.

!(#|2) is called a coherent upper conditional prevision and it is separately coherent if all the !(#|%) are separately coherent.



The model For every % ∈ 2 denote by s the Hausdorff dimension of the conditioning event B and by ℎ6 the Hausdorff s-dimensional outer measure.

ℎ6 is called the Hausdorff outer measure associated with the coherent upper conditional prevision !(#|%).





Theorem 2. Let L(B) be the class of all bounded random variables on B and let m be a 0-1-valued finitely additive, but not countably additive, probability on ℘(%) such that a different m is chosen for each B. Then for each % ∈ 2 the functional !(#|%) defined on L(B) by

!(#|%) = 89:(;) < #=ℎ6

; if 0 < ℎ6(%) < +∞

!(#|%) = !(#%) if ℎ6(%) = 0, +∞

is a coherent upper conditional prevision. (Theorem 2 , S. Doria, Annals of Operation Research, Vol. 195, pp.38-44, 2012)



The unconditional coherent upper prevision !(#|Ω) is obtained as a particular case when the conditioning event is Ω.

Coherent upper conditional probabilities !($|%) are obtained when only 0-1 valued random variables are considered.

Theorem 3 Let m be a 0-1-valued finitely additive, but not countably additive, probability on ℘(%) such that a different m is chosen for each B. Then for each % ∈ 2 the function !(#|%) defined on ℘(%) by



!($|%) = ℎ6($%)ℎ6(%) %& 0 < ℎ6(%) < +∞

and by

!($|%) = !($%) %& ℎ6(%) = 0, +∞ is a coherent upper conditional probability.



Main Results Let B be a set with positive and finite Hausdorff outer measure in its Hausdorff dimension. Denote by

';∗ ($) = !($|%) = ℎ6($%)ℎ6(%)

the coherent upper conditional probability defined on ℘(%).

From Theorem 2 we have that the coherent upper conditional prevision !(∙ |%) is a functional on *(%) with



values in ℝ and the coherent upper conditional probability ';∗ integral represents !(#|%) since

!(#|%) = , #=';∗ = 1

ℎ6(%) , #=ℎ6

;



If the conditioning event has positive and finite Hausdorff outer measure in its

Hausdorff dimension and K(B) is a linear lattice of bounded random variables

containing all constants

Necessary and sufficient conditions for a coherent upper conditional prevision to be

uniquely represented as the Choquet integral with respest to the upper conditional probability defined by its associated Hausdorff outer measure are to be :

Monotone Comonotonically additive Submodular Continuous from

below



Hausdorff outer measures Given a non-empty set Ω, let ℘(Ω) be the class of all subsets of Ω. An outer measure is a function '∗: ℘(Ω) → [0, +∞] such that '∗(∅) = 0, '∗($) ≤ '∗($2) if $ ⊆ $′ (monotonicity)

'∗ 5⋃ $7897:8 ; ≤ ∑ '∗($7)89

7:8 (subadditivity)



Given an additive set function ' defined on a class S properly contained in the power-set ℘(Ω) its outer measure '∗ and inner measure '∗ are defined on ℘(Ω) by

'∗($) = inf@'(%): % ⊃ $; % ∈ CD , $ ∈ ℘(Ω)

'∗($) = sup@'(%): % ⊂ $; % ∈ CD , $ ∈ ℘(Ω)



Examples of outer measures are the Hausdorff outer measures (named for Felix Hausdorff 1868 – 1942)


DOCTORAL COURSE IN EARTH SYSTEMS AND BUILT ENVIRONMENTS (Rogers C., 1970, Falconer K., 1986) Let (Ω,d) be a metric space.

Metric spaces Let Ω be a non-empty set and =: Ω × Ω → ℝ a function such that 1M) ∀ H, I ∈ Ω =(H, I) = 0 ⇔ H = I 2M)∀ H, I, K ∈ Ω =(H, I) + =(H, K) ≥ =(I, K) = is called a metric on Ω and the pair (Ω, =) is a metric space.

If in 2M) I = K, from 1M) we have



=(H, I) ≥ 0

If in 2M) H = K we have =(H, I) ≥ =(I, H).

Since it holds for every H, I ∈ Ω we have =(I, H) ≥ =(H, I)

That is =(H, I) = =(I, H).

A topology, called the metric topology, can be introduced in any metric space by defining the open sets of the space as the sets G with the property:



If x is a point of G, then for some r > 0 all points y with =(H, I) <M also belong to G.



Let (Ω,d) be a metric space. The diameter of a non - empty subset U of Ω is defined as

|N| = sup@=(H, I) ∶ H, I ∈ ND For δ > 0 the class @N7D7:8

89 is called a δ-cover of a subset E of Ω if

P ⊆ ⋃ N7897:8 and 0 < |N7| ≤ Q





Let s be a non-negative number. We define the function

ℎR6 (P) = inf S|N7|6

9

7:8

where the inferior is over all countable δ-covers @N7D7:889.



The Hausdorff s-dimensional outer measure of E, denoted by ℎ6, is defined as

ℎ6(P) = limR↦W ℎR6 (P)

This limit exists, but may be infinite, since ℎR6 increases as δ

decreases because less δ-covers are available.



Remark 1 : The limit of a monotone decreasing function when the variable δ goes to the inferior of its possible values exists and is equal to the supreme value of the possible values of the function. Remark 2: The function ℎR

6 is a decreasing function on δ,

For every Q and Q8 : Q < Q8 ⟹ ℎR6 (P) ≥ ℎR[

6 (P)

In fact if Q < Q8 the class of δ-covers is less than the class of Q8-covers because all the γ-covers with Q < \ < Q8 are Q8-covers but not δ-covers.



So for every subset E of Ω the function ℎR6 is a decreasing

function on δ since when δ decreases the inferior increases because it is calculated on a smaller class of δ-covers.

The Hausdorff dimension of a set A , dim](A), is defined as the unique value, such that

ℎ6($) = +∞ if 0 ≤ _ < dim]($)

ℎ6($) = 0 if dim]($) < _ < +∞



If 0 < ℎ6($) < +∞ then dim]($) = _, but the converse is not true (If A is a countable set → dim]($) = 0 and ℎW($) = +∞).

The Hausdorff dimension is the critical value of s at which the jump from +∞ to 0 occurs (Falconer 1990).



Example Let (Ω, =) be the Euclidean metric space with Ω = [0,1]`. Let P ⊆ Ω be a square of side l.



ℎR6 (P) = inf S|N7|6

9

7:8

|N7| = ab cde

`+ b c

de`

= cd √2 < δ

ℎR6 (P) = inf S|N7|6 = h` i j

h √2k69

7:8

ℎ6(P) = limR↦W

ℎR6 (P) = lim

d→9h` b c

d √2e6



_ = 0 ℎW(P) = limd→9

h` b cd √2e

W= +∞

_ = 1 ℎ8(P) = limd→9

h` b cd √2e

8= +∞

_ = 2 ℎ`(P) = limd→9

h` b cd √2e

`= 2j`

_ = 3 ℎm(P) = lim

d→9h` b c

d √2em

= 0



So, 0 < ℎ`(P) < +∞ ⟹ dim](P) = 2



A subset A of Ω is called measurable with respect to the outer measure ℎ6 if it decomposes every subset of Ω additively, that is

ℎ6(P) = ℎ6($ ∩ P) + ℎ6(P − $) for all sets P ⊆ Ω. The class of all measurable sets is a sigma-field.



Hausdorff s-dimensional outer measures are metric outer measures,

ℎ6(P ∪ p) = ℎ6(P) + ℎ6(p)

whenever =(P, p) = inf@=(H, I): H ∈ P, I ∈ pD > 0



All Borel subsets of Ω are measurable with respect to any metric outer measure (Falconer 1986, Theorem 1.5) So every Borel subset of Ω is measurable with respect to every Hausdorff outer measure ℎ6 since Hausdorff outer measures are metric.

The restriction of the Hausdorff outer measure ℎ6 to the sigma-field of ℎ6-measurable sets, containing the sigma-field of the Borel sets, is called Hausdorff s-dimensional measure.



The Hausdorff 0-dimensional measure is the counting measure; the Hausdorff 1-dimensional measure is the Lebeasgue measure

The Hausdorff s-dimensional measures are modular on the Borel sigma-field, that is

ℎ6($ ∪ %) + ℎ6($ ∩ %) = ℎ6($) + ℎ6(%)



for every pair of Borel sets A and B. So (Denneberg, 1994 Proposition 2.4) Hausdorff outer measures are submodular (or 2-alternating)

ℎ6($ ∪ %) + ℎ6($ ∩ %) ≤ ℎ6($) + ℎ6(%)



Denote by O the class of all open sets of Ω and by C the class of all compact sets of Ω, the restriction of each Hausdorff s-dimensional outer measure to the class H of all ℎ6-measurable sets with finite Hausdorff s-dimensional outer measure is strongly regular , that is it is regular outer regular (a) ℎ6($) = inf@ℎ6(N)|$ ⊂ N, N ∈ rD for all A∈ H inner regular (b) ℎ6($) = sup@ℎ6(s)|s ⊂ $, s ∈ tD for all A∈ H



With the additional property (c) inf @ℎ6(N − $)|$ ⊂ N, N ∈ rD = 0 for all A∈ H. Any Hausdorff s-dimensional outer measure is continuous from below (Falconer 1986, Lemma 1.3), that is for any increasing sequence u$vw of sets

limv→9

ℎ6x$vy = ℎ6 i limv→9

$vk



Any Hausdorff s-dimensional outer measure is translation invariant (Falconer 1986, p.18)

ℎ6(H + P) = ℎ6(P) where H + P = @H + I: I ∈ PD





Dr. Serena Doria

Department of Engineering and Geology, University G. d’Annunzio


Cohere Coherent conditional measures of risk

Coherent conditional measures of risk defined be the Choquet integral with respect to Hausdorff outer measures and dependent risks


Motivations

A new definition of coherent conditional

measure of risk based on Hausdorff outer

measures

A new definition of stochastic

independence for random variables


A new model of coherent upper conditional previsions based on Hausdorff outer measures has been introduced because conditional expectation defined in the axiomatic way , by the Radon-Nikodym derivative, may fail to be coherent. (Doria, 2012 Theorem 1)


A new model of coherent upper conditional prevision based on Hausdorff outer

measures (Doria, 2012 Theorem 2)

Let m be a 0-1 valued finitely additive, but not countably additive, probability on ℘(#) such that a different m is chosen for each #. Then for each # ∈ & the functionals '((|#) defined on *(#) by

'((|#) =,-. 1

ℎ1(#) 2 (3ℎ1 45 0 < ℎ1(#) < +∞#

:((#) 45 ℎ1(#) = 0, +∞

are separately coherent upper conditional previsions.


• The unconditional prevision is obtained when the conditioning event is Ω.

• '((|&)(=) is the random variable equal to '((|#) if = ∈ #.


Coherent upper conditional probabilities are obtained when only indicator functions of events are considered.

(Doria, 2012 Theorem 3)

Let m be a 0-1 valued finitely additive, but not countably additive, probability on ℘(#) such that a different m is chosen for each #. Then for each # ∈ & the function '(∙ |#) defined on ℘(#) by

'( |#) =,-

.ℎ1( #)ℎ1(#) 45 0 < ℎ1(#) < +∞

:( #) 45 ℎ1(#) = 0, +∞

is a coherent upper conditional probability.


A coherent conditional measure of risk !(∙ |#) is a functional on the class "(#) of all random variables (or risks) defined on B, such that the following axioms are satisfied for every (, # ∈ "(#):

(i) Monotonicity ( ≤ # ⇒ !((|#) ≤ !((|#)

(ii) Subadditivity !(( + #|#) ≤ !((|#) + !(#|#)

(iii) Traslation invariance !(( + ℎ|#) ≤ !((|#) + ℎ

(iv) Positive homogeneity !('(|#) = ' !((|#)


Let # ∈ & be a set with positive and finite Hausdorff outer measure in its Hausdorff dimension, i.e. 0 < ℎ1(#) < +∞, the functional !(∙ |#) defined on L(B) by

!((|#) = '((|#) = 1ℎ1(#) 2 (3ℎ1

(

is a coherent conditional measure of risk, which is comonotonically additive and continuous from below.


MOTIVATIONS

• In a continuous probabilistic space, where the probability is usually assumed equal to the Lebesgue measure, the standard definition of independence given by the factorization property

'( ∩ #) = '( )'(#)

is always satisfied for finite, countable and fractal sets (i.e. sets with non integer Hausdorff dimension) since both members of the equality are zero.


Hausdorff dimension of a set

Ω

DimH=2

DimH=1


Sierpinsky Triangle

DimH=

log3/log2


• If coherent upper and lower conditional probabilities are defined by outer and inner Hausdorff measures, an event B is always irrelevant, according to the definition proposed by Walley, to an event A if

dim-A < dim-B < dim-Ω. In fact the equalities

'( |#) = '( |#0) = '( |Ω)

'( |#) = '( |#0) = '( |Ω)

vanish to 0=0.

So the notion of s-irrelevance for events has been introduced (Doria 2007).


B is s-irrelevant to A if the following conditions hold

dim-A = dim-B = dim-AB

'( |#) = '( |#0) = '( |Ω)

'( |#) = '( |#0) = '( |Ω)

If B is s-irrelevant to A and A is s-irrelevant to B thus

A and B are s-independent


S-irrelevance and s-independence for random variables

The class of the weak upper level sets of the random variable X

1 = 23= ∈ Ω: ((=) ≥ 67 6 ∈ ℜ 7

Ω = [0,1]

A

X x

1


The atoms of the class 1 of the weak upper level sets of the random variable X are the minimal sets with respect to inclusion in 1 − ∅

1(() = 3 7

1(() = 23 7 ∪ 3#7>

X

B A


The atom is the union of the points where the random variable assumes the maximum value.

If a random variable X is such that the class of the upper level sets contains the n-Cantor sets ?@ where

?A = [0,1] , ?B = C0, BDE ∪ CF

D , 1E ,

?F = C0, BGE ∪ CF

G , BDE ∪ CF

D , HGE ∪ CI

G , 1E…….

? = ?JKLMN OPL = Q ?@

R

@SB

1(() = 3?7 34:T? = UVWFUVWD


This kind of structure of the upper level sets and of the atom describes stability in the random variable because we know that after a certain interval (of time) the random variable assumes again the maximum value.

DOCTO

RAL COURSE IN EARTH SYSTEM

S AND BUILT ENVIRONM

ENTS

Jan-38 M

ay-01

The blue graph is related to the rainfalls in S. Eufemia ( a m

ountain village near Pescara, Abruzzo, Italy))

The pink graph is related to the water flow

rate of the source Sorgente Verde (Abruzzo, Italy). Sorgente Verde has been a w

ater source not dry during this period (and also now

). (The author is grateful to Prof. Rusi -Dept. InGeo- for the data and the graph)

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5,0

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S-irrelevance and s-independent for random variables

Let X and Y two random variables and let S(X) and S(Y) respectively the classes of their atoms.

S(Y) is s-irrelevant to S(X) if for every E ∈ S(X) and F ∈ S(Y)

Y is s-irrelevant to X ⇔ S(Y) is s-irrelevant to S(X)

X and Y are s-independent ⇔ S(X) is s-irrelevant to S(Y) and S(Y) is s-irrelevant to S(X)


S-dependent random variables

1(() = 3 7 1(#) = 3#7

Ω = [0,1]

34:T = 34:T# = 0 ≠ 34:T #

X

B

Y

A


Since # = ∅

⇓

X and Y are s-dependent


S-independent random variables

Ω = [0,1]

3/4 1/2 1/4 1

B

X

Y

A


34:T = 34:T# = 34:T # = 1

'( |#) = '( |#[) = '( )

'(#| ) = '(#| 0) = '(#)

⇓

X and Y are s-independent


• can be represented as Choquet integral• are comonotonically additive • are countinuous from below

Coherent conditional measures of risk based on

Hausdorff outer measures

• s-independence of the indicator functions of two sets does not imply s-independence of the indicator functions of their complements

• s-independence for random variables does not imply the factorization of the joint distribution into the product of the marginal distributions

S-independence for random variables or

risks


According to the axiomatic definition two random variables X and Y are independent if and only if the sigma-fields generated by them, are independent. Since the sigma-field generated by a random variable X is the smallest sigma-field with respect to which X is measurable, it contains the inverse image of all borelian sets of ℜ. So if the random variables X and Y are independent then the joint distribution is equal to the product of the marginal distributions.


S-independence of random variables X and Y does not imply that the joint distribution is equal to the product of the marginal and . It occurs because s-independence between random variables implies that the factorization property holds only for the atoms (i.e. minimal sets with respect to the inclusion) of the classes generated by the weak upper level sets of X and Y and not for all sets of the sigma-fields generated by X and Y .




Dr. Serena Doria

Department of Engineering and Geology, University G. d’Annunzio

The aim of the course is to introduce some new results proposed in literature to represent and handling uncertainty in complex systems.



Iterated function systems

Let (ℜn, d) be the Euclidean metric space. A function f : ℜn → ℜn is called a contraction if

d( f (x), f (y)) ≤ rd(x, y) for all x, y ∈ ℜn, where 0 < r < 1 is a constant. The infimum value for which this inequality holds for all x, y is called the ratio of the contraction. A contraction that trasforms every subset of ℜn to a geometrically similar set is called a similitude. A similitude is a composition of a dilation, a rotation and translation. A set E is called invariant for a finite set of contractions f1, f2, ..., fm if

# = % &'

(

')*(#)



If the contractions are similitudes and for some s we have

hs(E) > 0 and

hs( fi(E) ∩ fj(E)) = 0 for - ≠ / then E is self similar. For any finite set of contractions there exists a unique non-empty compact invariant set K (Falconer, 1986, Theorem 8.3), called attractor. Given a finite set of contractions f1, f2, ..., fm we say that the open set condition (OSC) holds if there exists a bounded open set O such that

O ⊂⋃ &'(O) (')* and &'(O) ∩ &5(O) = ∅ for - ≠ /.

If f1, f2, ..., fm are similitudes with similarity ratios ri for i = 1...m the similarity dimension, which has the advantage of being easily calculable, is the unique positive number s for which



7(89): = ;<

9);

If the OSC holds then the compact invariant set K is self-similar and the Hausdorff dimension and the similarity dimension of K are equal.



The model of Sierpinski triangle

Let’s try to clarify what above said by introducing a classic

example of an attractor of a family of similarities known as

Sierpinski triangle.

Let = = >?, ;! × >?, ;! (the square in the figure) e let E be

the triangle of vertices O(0,0), A(1,0), B#;$ , ;%.

All points & ∈ = represent the possible states in which the

system is to be situated after the first meeting.

E



Let consider the following family of similarities '(9)9);* :

(;: ,- = ;

$ .

/ =;

; $ 0

($: ,- = ;

$ .

/ =;

; $ 0 + ;

$

2$: ,- = ;

$ . + ;$

/ =;; $ 0 + ;

$

.

Iteratively applying the three transformations to the initial

triangle E you get the attractor of the system of similarities,

known as the Sierpinsky triangle shown in the following

figure:



The Hausdorff dimension of the Sierpinski triangle coincides

with the size of the similarity dimension s and is calculated

by solving the equation (1):

* 3;$4

:= ;

that is : = 567$567* < $ .

The Sierpinski triangle is a fractal, because its Hausdorff

dimension has a fractional number, and is self-similar

because its geometric structure is repeated identically at

different scales. This property is also expressed by saying



that a single part reproduces the whole. In addition, the

Sierpinski triangle presents symmetries.

Fractal Antennas

Since fractals show up in the real world of nature (snail shells, leaves on a tree, pine cones), why not see if they perform well as antennas? It turns out that if you make antennas with fractal shapes, they will radiate, and often have multiband properties. Exmaples of these antennas are below:





s-Independence for curves filling the space The notions of s-irrelevance and s-independence for events A and B that are represented by curves filling the space are analyzed. In particular Peano curve, Hilbert curve and Peano-Sierpinski curve are proven to be s-independent. Curves filling the space Sagan (1994) can be defined as the limit of a Cauchy sequence of continuous functions fn, each mapping the unit interval into the unit square. The convergence is uniform so that the limit is a continuous function, i.e. a curve.

Six iterations of the Hilbert curve construction, whose limiting space-filling curve was devised by

mathematician David Hilbert.



Definition Let (Ω, d) be a metric space and let A and B be two curves filling the space _. Then A and B are s-independent if the following conditions hold • 1s) dimH(AB) = dimH(B)= dimH(A) • 2s) P(A|B) = P(A) and P(A|B) = P(A); • 3s) P(B|A) = P(B) and P(B|A) = P(B); Moreover B is s-irrelevant to A if conditions 1s) and 2s) are satisfied. Theorem Let Ω = [0, 1]n and let :; and : be the upper and lower conditional probabilities defined as in Theorem 4. If A and B are two curves filling the space then A and B are s-independent.

Documents

Handling Uncertainty in Complex Systems · Handling Uncertainty in Complex Systems Dr. Serena Doria Department of Engineering and Geology, University G. d’Annunzio The aim of the